I have a strawberry birthmark on my leg which has started bleeding. Should I get it checked?

Yes. "strawberry" birthmarks are actually arteriovenous malformations or simply venous malformations. Some of these have fairly large veins under them. It wouls be best to see a physican that is well versed in looking at this type of problem.
Yes. Strawberry hemangiomas are made of blood vessels, so bleeding is not a good thing. Lasers are available which shrink down hemangiomas. See a dermatologist who has experience with vascular lasers.